Portfolio optimization with unobservable Markov-modulated drift process

We study portfolio optimization problems in which the drift rate of the stock is Markov modulated and the driving factors cannot be observed by the investor. Using results from filter theory, we reduce this problem to one with complete observation. In the cases of logarithmic and power utility, we solve the problem explicitly with the help of stochastic control methods. It turns out that the value function is a classical solution of the corresponding Hamilton-Jacobi-Bellman equation. As a special case, we investigate the so-called Bayesian case, i.e. where the drift rate is unknown but does not change over time. In this case, we prove a number of interesting properties of the optimal portfolio strategy. In particular, using the likelihood-ratio ordering, we can compare the optimal investment in the case of observable drift rate to that in the case of unobservable drift rate. Thus, we also obtain the sign of the drift risk.

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Optimal Portfolios for Financial Markets with Wishart Volatility

Journal of Applied Probability ◽

10.1239/jap/1389370097 ◽

2013 ◽

Vol 50 (4) ◽

pp. 1025-1043 ◽

Cited By ~ 14

Author(s):

Nicole Bäuerle ◽

Zejing Li

Keyword(s):

Value Function ◽

Numerical Study ◽

Heston Model ◽

Hard Part ◽

Control Approach ◽

Portfolio Strategy ◽

Hamilton Jacobi Bellman ◽

The One ◽

Hedging Demand ◽

The Value Function

We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain, explicitly, the optimal portfolio strategy and the value function in some parameter settings. In particular, we do this when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is to identify when the solution of the Hamilton-Jacobi-Bellman equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.

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An Investment and Consumption Problem with CIR Interest Rate and Stochastic Volatility

Abstract and Applied Analysis ◽

10.1155/2013/219397 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 7

Author(s):

Hao Chang ◽

Xi-min Rong

Keyword(s):

Interest Rate ◽

Stochastic Volatility ◽

Optimal Investment ◽

Stochastic Volatility Model ◽

Power Utility ◽

Variable Approach ◽

Volatility Model ◽

Stochastic Interest ◽

Hamilton Jacobi Bellman ◽

Optimal Investment And Consumption

We are concerned with an investment and consumption problem with stochastic interest rate and stochastic volatility, in which interest rate dynamic is described by the Cox-Ingersoll-Ross (CIR) model and the volatility of the stock is driven by Heston’s stochastic volatility model. We apply stochastic optimal control theory to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function and choose power utility and logarithm utility for our analysis. By using separate variable approach and variable change technique, we obtain the closed-form expressions of the optimal investment and consumption strategy. A numerical example is given to illustrate our results and to analyze the effect of market parameters on the optimal investment and consumption strategies.

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OPTIMAL INVESTMENT AND REINSURANCE IN A JUMP DIFFUSION RISK MODEL

The ANZIAM Journal ◽

10.1017/s144618111100068x ◽

2011 ◽

Vol 52 (3) ◽

pp. 250-262 ◽

Cited By ~ 10

Author(s):

XIANG LIN ◽

PENG YANG

Keyword(s):

Risk Model ◽

Optimal Investment ◽

Jump Diffusion ◽

Risk Process ◽

Proportional Reinsurance ◽

Insurance Company ◽

Closed Form Solutions ◽

Hamilton Jacobi Bellman Equation ◽

Hamilton Jacobi Bellman ◽

The Value Function

AbstractWe consider an insurance company whose surplus is governed by a jump diffusion risk process. The insurance company can purchase proportional reinsurance for claims and invest its surplus in a risk-free asset and a risky asset whose return follows a jump diffusion process. Our main goal is to find an optimal investment and proportional reinsurance policy which maximizes the expected exponential utility of the terminal wealth. By solving the corresponding Hamilton–Jacobi–Bellman equation, closed-form solutions for the value function as well as the optimal investment and proportional reinsurance policy are obtained. We also discuss the effects of parameters on the optimal investment and proportional reinsurance policy by numerical calculations.

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Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model

Risks ◽

10.3390/risks9010017 ◽

2021 ◽

Vol 9 (1) ◽

pp. 17

Author(s):

Leonie Violetta Brinker

Keyword(s):

Brownian Motion ◽

Optimal Investment ◽

Investment Strategy ◽

Insurance Company ◽

Piecewise Monotone ◽

Expected Time ◽

Optimal Investment Strategy ◽

Running Maximum ◽

Hamilton Jacobi Bellman ◽

The Value Function

Consider an insurance company whose surplus is modelled by an arithmetic Brownian motion of not necessarily positive drift. Additionally, the insurer has the possibility to invest in a stock modelled by a geometric Brownian motion independent of the surplus. Our key variable is the (absolute) drawdown Δ of the surplus X, defined as the distance to its running maximum X¯. Large, long-lasting drawdowns are unfavourable for the insurance company. We consider the stochastic optimisation problem of minimising the expected time that the drawdown is larger than a positive critical value (weighted by a discounting factor) under investment. A fixed-point argument is used to show that the value function is the unique solution to the Hamilton–Jacobi–Bellman equation related to the problem. It turns out that the optimal investment strategy is given by a piecewise monotone and continuously differentiable function of the current drawdown. Several numerical examples illustrate our findings.

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On the Stochastic Optimal Control Model of the Investments of Defined Contribution (DC) Pension Funds

Journal of Applied Sciences and Environmental Management ◽

10.4314/jasem.v24i2.12 ◽

2020 ◽

Vol 24 (2) ◽

pp. 63-269

Author(s):

T. Latunde ◽

O.O. Esan ◽

J.O. Richard ◽

D.D. Dare

Keyword(s):

Sensitivity Analysis ◽

Optimization Problems ◽

Absolute Risk ◽

Optimal Investment ◽

Investment Strategy ◽

Pension Funds ◽

Model Parameters ◽

Defined Contribution ◽

Risky Assets ◽

Hamilton Jacobi Bellman

One of the major problems faced in the management of pension funds and plan is how to allocate and control the future flow of contribution likewise the proportion of portfolio value and investments in risky assets. In this work, optimal investment for a stochastic model of a Defined contribution (DC) is investigated such that the model design is analysed yielding an optimized expected utility of the members’ terminal wealth. An optimized solution is derived using the Hamilton Jacobi equation in solving the problem of investment strategy formulated by Constant absolute risk aversion (CARA). However, to consider the changes that occur in the dimension of optimal solutions in optimization problems, mostly, the optimal behaviour of parameters, the sensitivity analysis is considered. Thus, the analysis of the model is carried out herein by utilising the approach of the sensitivity analysis of parameters. This is carried out by using Maple software and varying the values of some model parameters such that the behaviour of each parameter relating to the pension funds invested in the risky assets is determined. The results are presented graphically and using tables thus discussed such that pension investors and stakeholders are advised. Keywords: Stochastic; DC Pension funds; Sensitivity analysis; Hamilton-Jacobi-Bellman equation; Optimal investment

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Optimal Portfolios for Financial Markets with Wishart Volatility

Journal of Applied Probability ◽

10.1017/s0021900200013772 ◽

2013 ◽

Vol 50 (04) ◽

pp. 1025-1043 ◽

Cited By ~ 9

Author(s):

Nicole Bäuerle ◽

Zejing Li

Keyword(s):

Value Function ◽

Numerical Study ◽

Heston Model ◽

Hard Part ◽

Control Approach ◽

Portfolio Strategy ◽

Hamilton Jacobi Bellman ◽

The One ◽

Hedging Demand ◽

The Value Function

We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain, explicitly, the optimal portfolio strategy and the value function in some parameter settings. In particular, we do this when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is to identify when the solution of the Hamilton-Jacobi-Bellman equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.

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Optimal Investment-Consumption Strategy under Inflation in a Markovian Regime-Switching Market

Discrete Dynamics in Nature and Society ◽

10.1155/2016/9606497 ◽

2016 ◽

Vol 2016 ◽

pp. 1-17 ◽

Cited By ~ 1

Author(s):

Huiling Wu

Keyword(s):

Regime Switching ◽

Stock Price ◽

Optimal Investment ◽

Investment Strategy ◽

Power Utility ◽

Optimal Investment Strategy ◽

Admissible Strategies ◽

Definition Of ◽

Markovian Regime Switching ◽

Optimal Investment And Consumption

This paper studies an investment-consumption problem under inflation. The consumption price level, the prices of the available assets, and the coefficient of the power utility are assumed to be sensitive to the states of underlying economy modulated by a continuous-time Markovian chain. The definition of admissible strategies and the verification theory corresponding to this stochastic control problem are presented. The analytical expression of the optimal investment strategy is derived. The existence, boundedness, and feasibility of the optimal consumption are proven. Finally, we analyze in detail by mathematical and numerical analysis how the risk aversion, the correlation coefficient between the inflation and the stock price, the inflation parameters, and the coefficient of utility affect the optimal investment and consumption strategy.

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The Optimal Strategy of Reinsurance-Investment Problem for an Insurer with Dynamic Income Under Stochastic Interest Rate

Journal of Systems Science and Information ◽

10.21078/jssi-2016-244-14 ◽

2016 ◽

Vol 4 (3) ◽

pp. 244-257

Author(s):

Delei Sheng

Keyword(s):

Interest Rate ◽

Interest Rates ◽

Control Technique ◽

Stochastic Interest Rate ◽

Insurance Company ◽

Power Utility ◽

Investment Problem ◽

Stochastic Interest ◽

Hamilton Jacobi Bellman ◽

Constant Interest Rate

AbstractThis paper considers the reinsurance-investment problem for an insurer with dynamic income to balance the profit of insurance company and policy-holders. The insurer’s dynamic income is given by a net premium minus a dynamic reward budget item and the net premium is obtained according to the expected premium principle. Applying the stochastic control technique, a Hamilton-Jacobi-Bellman equation is established under stochastic interest rate model and the explicit solution is obtained by maximizing the insurer’s power utility of terminal wealth. In addition, the comparison with corresponding results under constant interest rate helps us to understand the role and influence of stochastic interest rates more in-depth.

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Optimal Investment and Consumption for Multidimensional Spread Financial Markets with Logarithmic Utility

Stats ◽

10.3390/stats4040058 ◽

2021 ◽

Vol 4 (4) ◽

pp. 1012-1026

Author(s):

Sahar Albosaily ◽

Serguei Pergamenchtchikov

Keyword(s):

Financial Markets ◽

Numerical Simulations ◽

Financial Market ◽

Optimal Investment ◽

Optimal Consumption ◽

Dynamical Programming ◽

Verification Theorem ◽

Logarithmic Utility ◽

Hamilton Jacobi Bellman ◽

Optimal Investment And Consumption

We consider a spread financial market defined by the multidimensional Ornstein–Uhlenbeck (OU) process. We study the optimal consumption/investment problem for logarithmic utility functions using a stochastic dynamical programming method. We show a special verification theorem for this case. We find the solution to the Hamilton–Jacobi–Bellman (HJB) equation in explicit form and as a consequence we construct optimal financial strategies. Moreover, we study the constructed strategies with numerical simulations.

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